Paper folding

A rectangular, flat sheet of paper, 30 by 20cm is placed on a flat surface. One of the vertices is lifted up and placed on the opposite edge, while the other three vertices are held in their original positions. With all four vertices now held fixed, the paper is smoothed flat as shown in the diagram. Note that AR = RQ, AP = PQ.

Let the length of the crease, PR, be L cm and AP be x cm. Express L as a function of x. Using a spreadsheet, graphics calculator or table of values, draw a graph of this function. Hence, determine the minimum length of the crease, and the size of angle PQB for the minimum length

Trigonometry

Hello m_i_k_o

Originally Posted by m_i_k_o

A rectangular, flat sheet of paper, 30 by 20cm is placed on a flat surface. One of the vertices is lifted up and placed on the opposite edge, while the other three vertices are held in their original positions. With all four vertices now held fixed, the paper is smoothed flat as shown in the diagram. Note that AR = RQ, AP = PQ.

Let the length of the crease, PR, be L cm and AP be x cm. Express L as a function of x. Using a spreadsheet, graphics calculator or table of values, draw a graph of this function. Hence, determine the minimum length of the crease, and the size of angle PQB for the minimum length

Let , where is measured in degrees.

Then

So, from

So, using

Now (I assume from your diagram that is the shorter side of the rectangle)

This, then, is as a function of . Plotting this in Excel, I produced the chart as shown. This gives a minimum value of as cm (1 d.p.) when . For this value of